Calculus Of One Real Variable
– By Pheng Kim Ving

5.5 
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1. The SecondDerivative Test 
In Section
5.3 Theorem 4.1 we had the firstderivative test for local extrema. In this
section we're
going to study the
secondderivative test, which is also a test for local extrema.
Let f
be a function that is twice differentiable near x_{1} and also near x_{2}. See Fig. 1.1. Suppose f
'(x_{1}) = 0 and f
''(x_{1}) > 0.
So the graph of f has a horizontal tangent at x_{1} and is concave up there. This means that f(x_{1}) is a local minimum
of f.
Similarly, if f '(x_{2}) = 0 and f ''(x_{2}) < 0, then f(x_{2}) is a local maximum of f.
Fig. 1.1
f(x_{1}) is a local minimum and f(x_{2}) is a local maximum of f.

Theorem 1.1 – The SecondDerivative Test
i. If f '(x_{1}) = 0 and f ''(x_{1}) > 0, then f has a local
minimum at x_{1}. 
Proof
We prove part i. The proof
of part ii is similar. Recalling that f
'' is the derivative of f '
and using the definition of the
derivative we have:
see note on the proof, below. Consider such small h's. If h < 0 then
f '(x_{1} + h) < 0, and if h
> 0 then f '(x_{1} + h) > 0.
Thus, by the firstderivative test, f has a local
minimum at x_{1}.
Because h – 0 = h and f ''(x_{1})/2 > 0, the justification is complete.
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2. A Closer Look At The SecondDerivative Test 
i. The secondderivative test is a test for
local extrema, not for inflection points. It has the same purpose as the
firstderivative
test.
ii. The secondderivative test excludes points x_{1} where f '(x_{1}) doesn't
exist. As f '(x_{1}) doesn't
exist, f ''(x_{1}) doesn't
exist
either
(if a function ( f ' in this
case) isn't
defined at x_{1}, then it can't be differentiable there). So it's meaningless to
talk
about
the second derivative of f at x_{1}.
iii. The secondderivative test asserts nothing about what happens at points x_{1} where f '(x_{1}) = 0 and f ''(x_{1}) = 0 too.
Let's take a look at some examples. The
graph of f_{1}(x) = x^{4} is sketched in Fig. 2.1. We have f_{1}'(0)
= f_{1}'(x)_{x}_{=}_{0} =
4x^{3}_{x}_{=}_{0} = 0 and f_{1}''(0) = f_{1}''(x)_{x}_{=}_{0} = 12x^{2}_{x}_{=}_{0} = 0. Clearly f_{1} has a local minimum at x
= 0. The graph of f_{2}(x)
= –x^{4} is
sketched in Fig. 2.2. We have f_{2}'(0)
= f_{2}'(x)_{x}_{=}_{0} = – 4x^{3}_{x}_{=}_{0} = 0 and f_{2}''(0) = f_{2}''(x)_{x}_{=}_{0} = –12x^{2}_{x}_{=}_{0} = 0. Clearly f_{2} has a
local maximum at x = 0. The graph of f_{3}(x) = x^{3} is sketched in Fig. 2.3. We have f_{3}'(0)
= f_{3}'(x)_{x}_{=}_{0} = 3x^{2}_{x}_{=}_{0} = 0 and
f_{3}''(0)
= f_{3}''(x)_{x}_{=}_{0} = 6x_{x}_{=}_{0} = 0. Clearly f_{3} has neither a local maximum nor a local minimum at x = 0, but it has an
inflection point there. Thus, if f '(x_{1}) = 0 and f ''(x_{1}) = 0, then f can have
either a local maximum or a local minimum
or an inflection point at x_{1}.
Fig. 2.1 f_{1}(x) = x^{4}; 
Fig. 2.2 f_{2}(x) = – x^{4}; 
Fig. 2.3 f_{3}(x) = x^{3}; 
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3. Using The SecondDerivative Test 
Example 3.1
Use the secondderivative test to find
and classify the critical points of f(x) = x – (1/2)x^{2} –xe^{–}^{x}. (The derivative of e^{x} with
respect to x is e^{x}
itself.)
Solution
f '(x) = 1 – x – e^{–x} + xe^{–x} = 1 – x – e^{–x}(1 – x) = (1 – x)(1 – e^{–x});
f '(x) is defined
everywhere;
f '(x) = 0 iff (1
– x)(1 – e^{–x}) = 0 iff (1 – x) = 0 or (1
– e^{–x})
= 0, so f '(x) = 0 at x = 1 and x = 0.
f ''(x) = – 1 + e^{–x} + e^{–x} – xe^{–x} = – 1 + e^{–x}(2 – x);
f ''(1)
= – 1 + e^{–1}(2 – 1) = – 1 + 1/e < – 1 + 1 = 0;
f ''(0)
= – 1 + 1(2 – 0) = 1 > 0.
So f has a local maximum at x = 1 and a local minimum at x = 0.
EOS
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4. Comparing The Second To The FirstDerivative Tests 
i. The secondderivative test is simpler than the firstderivative test if the second derivative isn't too complicated.
ii. However, the second derivative may be so
much more complicated than the first derivative that it may be less difficult
to use the firstderivative test
than the secondderivative test.
iii. The secondderivative test can't locate local
extreme values at singular points ( points where the derivative doesn't
exist), while the firstderivative
test can.
Problems & Solutions 
1. Let f(x) = x/(1 + x^{2}). Find and classify all local extrema of f using the secondderivative test.
Solution
So f(1) = 1/2 is a local maximum and f(–1) = –1/2 is a local minimum of f.
2. Let g(x) = x^{2}e^{–}^{x}.
Find and classify all local extrema of g using the
secondderivative test. (The derivative of e^{x}
with
respect to x
is e^{x} itself.)
g'(x) = 2xe^{–}^{x}
+ x^{2}(–e^{–}^{x})
= xe^{–}^{x}(2 – x);
g'(x) exists
everywhere;
g'(x) = 0 at x = 0 and x = 2;
g''(x) = (e^{–}^{x} + x(–e^{–}^{x}))(2 – x) + xe^{–}^{x}(–1)
= e^{–}^{x}(2 – 4x + x^{2});
g''(0) = 2 > 0, g''(2)
= –2e^{–2} < 0.
Thus by the secondderivative test g(0) = 0 is a local minimum and g(2) = 4e^{–2} is a local maximum of g.
3. Let h(t) = t ln t. Find and classify all local
extrema of h using the secondderivative test.
(The derivative of ln x with
respect to x
is 1/x.)
h'(t) = ln t + t(1/t) = ln t + 1;
h'(t) is defined for all t
> 0;
h'(t) = 0 iff ln
t = –1 iff t = e^{–1} = 1/e;
h''(t) = 1/t;
h''(1/e) = e > 0.
Consequently by the secondderivative test h(1/e) = (1/e) ln
(1/e) = –1/e
is a local minimum of h.
4. The graph of the derivative f ' of a function f is shown below.
Let G be the graph of f (not of f '). The secondderivative test is handy for some of the following questions.
a. Does G has a local
maximum at x
= 0?
b. Does G
has a local maximum at x
= –1?
c. Does G
has a local minimum at x
= 1?
d. Does G has a local minimum at x = 2?
e. Does G
has an inflection point at x
= 0?
f. Does G
has an inflection point x
= 1?
g. Is G
concave up on (0, 2)?
h. Is G concave up on (1, 2)?
Solution
a. Yes.
b. No.
c. No.
d. Yes.
e. No.
f. Yes.
g. No.
h. Yes.
5. Give an example of three functions f, g, and h such that f '(0) = f ''(0) = g'(0) = g''(0) = h'(0) = h''(0)
= 0, but f
has
a local minimum value, g has a local
maximum value, and h
has an inflection point at x
= 0.
Solution
Let f(x) = x^{6}, g(x) = –x^{6}, and h(x) = x^{5}.
Note
For f and g, the
secondderivative test doesn't
apply but the firstderivative test does. For h, neither the first nor the
secondderivative test applies; so we check to see if the given point is an
inflection point.
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